Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > lmodvsass | GIF version | ||
| Description: Associative law for scalar product. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| Ref | Expression |
|---|---|
| lmodvsass.v | β’ π = (Baseβπ) |
| lmodvsass.f | β’ πΉ = (Scalarβπ) |
| lmodvsass.s | β’ Β· = ( Β·π βπ) |
| lmodvsass.k | β’ πΎ = (BaseβπΉ) |
| lmodvsass.t | β’ Γ = (.rβπΉ) |
| Ref | Expression |
|---|---|
| lmodvsass | β’ ((π β LMod β§ (π β πΎ β§ π β πΎ β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmodvsass.v | . . . . . . 7 β’ π = (Baseβπ) | |
| 2 | eqid 2177 | . . . . . . 7 β’ (+gβπ) = (+gβπ) | |
| 3 | lmodvsass.s | . . . . . . 7 β’ Β· = ( Β·π βπ) | |
| 4 | lmodvsass.f | . . . . . . 7 β’ πΉ = (Scalarβπ) | |
| 5 | lmodvsass.k | . . . . . . 7 β’ πΎ = (BaseβπΉ) | |
| 6 | eqid 2177 | . . . . . . 7 β’ (+gβπΉ) = (+gβπΉ) | |
| 7 | lmodvsass.t | . . . . . . 7 β’ Γ = (.rβπΉ) | |
| 8 | eqid 2177 | . . . . . . 7 β’ (1rβπΉ) = (1rβπΉ) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 13382 | . . . . . 6 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β (((π Β· π) β π β§ (π Β· (π(+gβπ)π)) = ((π Β· π)(+gβπ)(π Β· π)) β§ ((π(+gβπΉ)π ) Β· π) = ((π Β· π)(+gβπ)(π Β· π))) β§ (((π Γ π ) Β· π) = (π Β· (π Β· π)) β§ ((1rβπΉ) Β· π) = π))) |
| 10 | 9 | simprld 530 | . . . . 5 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
| 11 | 10 | 3expa 1203 | . . . 4 β’ (((π β LMod β§ (π β πΎ β§ π β πΎ)) β§ (π β π β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
| 12 | 11 | anabsan2 584 | . . 3 β’ (((π β LMod β§ (π β πΎ β§ π β πΎ)) β§ π β π) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
| 13 | 12 | exp42 371 | . 2 β’ (π β LMod β (π β πΎ β (π β πΎ β (π β π β ((π Γ π ) Β· π) = (π Β· (π Β· π)))))) |
| 14 | 13 | 3imp2 1222 | 1 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5217 (class class class)co 5875 Basecbs 12462 +gcplusg 12536 .rcmulr 12537 Scalarcsca 12539 Β·π cvsca 12540 1rcur 13142 LModclmod 13377 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-cnex 7902 ax-resscn 7903 ax-1re 7905 ax-addrcl 7908 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-ov 5878 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-5 8981 df-6 8982 df-ndx 12465 df-slot 12466 df-base 12468 df-plusg 12549 df-mulr 12550 df-sca 12552 df-vsca 12553 df-lmod 13379 |
| This theorem is referenced by: lmodvs0 13412 lmodvsneg 13421 lmodsubvs 13433 lmodsubdi 13434 lmodsubdir 13435 |
| Copyright terms: Public domain | W3C validator |